What I'm wondering is can I simply transfer the BTC from my Electrum into my Ledger Nano S (instead of transferring to a new Electrum address)
Yes.
and then install Electron Cash (on a different machine, although, post-successful-transfer, I doubt it matters, but better be safe than sorry) and use the old Electrum BTC address to claim BCH from?
You can, but you risk sending your BSV to the same address (on a different chain) too.
Just do not use your old seeds or private keys after claiming BCH from Electrum. Because you are going to input your seed/private keys to another wallet which is a fork of Electrum.
Exactly. You should consider your private keys compromised after doing this. This is also how several people lost their Bitcoin, and that's why you should always transfer them before doing anything else.
I don't want to rush, in case the exact sequence of doing things in order to claim fork coins is important.
I'd go from most valuable to least valuable coin, in that order. This way, if one of the wallets turns out to be compromised, you limit your losses.
And regarding BSV, which I'm just reading about... I understand there was a BCH hard fork about two weeks ago. Can I claim BSV too, or do I have to have claimed BCH FIRST, before 15th November? Or is it enough to have owned BTC prior to 15th November, regardless of whether I claimed BCH or not?
You own all these Forkcoins if you have "old" Bitcoins. BSV is not the recent fork, BSV came out of BCH, and BCH recently forked again (into
BCHABC. And none of them have replay protection!
The easiest way to split them is by receiving a small amount of each coin, and including that in the transaction.
In short:
1. Secure your Bitcoin by sending to your Ledger.
2. Receive $0.01 worth of BCH from an input that doesn't exist on any of the other chains. Include that $0.01 worth of BCH into your transaction, and it can't be replayed on other chains.
3. Do the same for BSV.
4. Do it for whatever new fork they make too.
There's also Bitcoin Gold and Bitcoin Diamond. I don't think they have replay problems, and depending how many Bitcoins you're holding it may or may not be worth your time. I haven't figured out the many other Forks yet, but I doubt it's still worth it.